Geometry of vector bundles and moduli spaces of instantons

本研究计划关注的是理解具有特殊结构的向量丛的各种模空间的拓扑和几何。 这样一个程序存在于微分几何的数学领域，流形上的微积分和微分方程的研究，更具体地说，它存在于规范理论的领域，向量丛和连接的研究。在数学和物理界一直在努力对杨-米尔斯瞬子进行分类，杨-米尔斯瞬子是曲率使一定能量最小化的连接。 这个研究项目的重点是在某些非紧流形上进行这种分类，也就是说，获得关于生活在这些空间上的瞬子空间的几何和拓扑的知识，模自然等价。理想情况下，人们可以用更容易处理的方程（常微分方程、代数方程）的解来表达这个空间，一个困难的偏微分方程的解的空间。 然后通过这样一个透镜来描述起始空间的几何形状。 在非紧凑型基础管汇上工作会产生严重的困难，必须开发工具来解决这些困难。*克服这些困难并提供关于这些模空间的信息可以更好地理解瞬子的物理学，提供关于底层流形的有价值的信息，并且本身就是有趣的数学。

This research program is concerned with understanding the topology and geometry of various moduli spaces of vector bundles equipped with special structures. Such a program lies in the mathematical field of differential geometry, the study of calculus and differential equations on manifolds, and more specifically it lies in the field of gauge theory, the study of vector bundles and connections.****There have been ongoing efforts in the mathematics and physics worlds to classify Yang-Mills instantons, which are connections whose curvature minimizes a certain energy. The focus of this research program is to do this classification on certain non-compact manifolds, that is to gain knowledge about the geometry and topology of the spaces of instantons living on those spaces, modulo natural equivalences. Ideally, one could express this space, a space of solutions to a difficult partial differential equation, in terms of solutions to more tractable equations (ordinary differential equation, algebraic equation). One then describes the geometry of the starting space through such a lens. Working on a non-compact base manifold creates serious difficulties and tools must be developed to address them.****Overcoming these difficulties and providing information about these moduli spaces allows a better understanding of the physics of instantons, provides valuable information about the underlying manifolds, and is interesting math on its own sake.**********************